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Chapter 1.1
Use mathematical tools to measure and predict.
Apply accuracy and precision when measuring.
Display and evaluate data graphically.
Chapter
1
In this chapter you will:
Mathematics and Physics
Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are related.
What is Physics?
Section
1.1
Learning physics will help you to understand the physical world.
Mathematics and Physics
Physics uses mathematics as a powerful language.
In physics, equations are important tools for modeling observations and for making predictions.
Mathematics in Physics
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Mathematics and Physics
The Système International d’Unités, or SI, uses seven base quantities, which are shown in the table below.
SI Units
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Mathematics and Physics
The base quantities were originally defined in terms of direct measurements. Other units, called derived units, are created by combining the base units in various ways.
The SI system is regulated by the International Bureau of Weights and Measures in Sèvres, France.
This bureau and the National Institute of Science and Technology (NIST) in Gaithersburg, Maryland, keep the standards of length, time, and mass against which our metersticks, clocks, and balances are calibrated.
SI Units
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Mathematics and Physics
Prefixes are used to change SI units by powers of 10, as shown in the table below.
SI Units
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To convert between SI units, multiply or divide by the appropriate power of 10.
Mathematics and Physics
You often will need to use different versions of a formula, or use a string of formulas, to solve a physics problem.
To check that you have set up a problem correctly, write the equation or set of equations you plan to use with the appropriate units.
Dimensional Analysis
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1.1
Mathematics and Physics
Before performing calculations, check that the answer will be in the expected units.
For example, if you are finding a speed and you see that your answer will be measured in s/m or m/s2, you know you have made an error in setting up the problem.
This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis.
Dimensional Analysis
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Mathematics and Physics
Dimensional analysis also is used in choosing conversion factors.
A conversion factor is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:
Dimensional Analysis
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g 1000kg 11
kg 1g 10001
Mathematics and Physics
Choose a conversion factor that will make the units cancel, leaving the answer in the correct units.
For example, to convert 1.34 kg of iron ore to grams, do as shown below:
Dimensional Analysis
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1.1
kg 1
g 1000kg 1.34 1340 g
Mathematics and Physics
A meterstick is used to measure a pen and the measurement is recorded as 14.3 cm.
This measurement has three valid digits: two you are sure of, and one you estimated.
The valid digits in a measurement are called significant digits.
Significant Digits
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However, the last digit given for any measurement is the uncertain digit.
Mathematics and Physics
All nonzero digits in a measurement are significant, but not all zeros are significant.
Consider a measurement such as 0.0860 m. Here the first two zeros serve only to locate the decimal point and are not significant.
The last zero, however, is the estimated digit and is significant.
Significant Digits
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1.1
Mathematics and Physics
When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least-precise measurement.
To add or subtract measurements, first perform the operation, then round off the result to correspond to the least-precise value involved.
To multiply or divide measurements, perform the calculation and then round to the same number of significant digits as the least-precise measurement.
Note that significant digits are considered only when calculating with measurements.
Significant Digits
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Section Check
The potential energy, PE, of a body of mass, m, raised to a height, h, is expressed mathematically as PE = mgh, where g is the gravitational constant. If m is measured in kg, g in m/s2, h in m, and PE in joules, then what is 1 joule described in base unit?
Question 1
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A. 1 kg·m/s
B. 1 kg·m/s2
C. 1 kg·m2/s
D. 1 kg·m2/s2
Section Check
Answer: D
Answer 1
Reason:
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1.1
2m
1J = 1kgs
2
m1J = 1kg m
s
PE mgh
Section Check
A car is moving at a speed of 90 km/h. What is the speed of the car in m/s? (Hint: Use Dimensional Analysis)
Question 2
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A. 2.5×101 m/s
B. 1.5×103 m/s
C. 2.5 m/s
D. 1.5×102 m/s
Section Check
Answer: A
Answer 2
Reason:
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90 km 1000 m 1 h 1 min 25 m/s1 h 1 km 60 min 60 s
Section Check
Which of the following representations is correct when you solve 0.030 kg + 3333 g using scientific notation?
Question 3
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A. 3.4×103 g
B. 3.36×103 g
C. 3×103 g
D. 3363 g
Section Check
Answer: A
Answer 3
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Reason: 0.030 kg can be written as 3.0 101 g which has 2 significant digits, the number 3 and the zero after 3.
In number 3333 all the four 3’s are significant hence it has 4 significant digits. So our answer should contain 2 significant digits.
